6.2
The Behavior of
2
×
2
Linear Systems
Now that we have seen an example showing the significance of the eigenvalues and eigenvectors
in understanding the behavior of the solutions to a linear system of di
ff
erential equations, let’s
systematically explore the 2
×
2 systems. Let
A
=
a
1
,
1
a
1
,
2
a
2
,
1
a
2
,
2
.
Then
det (
A

λ
I
) = det
a
1
,
1

λ
a
1
,
2
a
2
,
1
a
2
,
2

λ
=
λ
2

(
a
1
,
1
+
a
2
,
2
)
λ
+ (
a
1
,
1
a
2
,
2

a
1
,
2
a
2
,
1
)
.
(6)
We call the sum of the diagonal entries of a matrix,
M
, the
trace
and denote it by tr(
M
). Since
tr(
A
) = (
a
1
,
1
+
a
2
,
2
) and det(
A
) = (
a
1
,
1
a
2
,
2

a
1
,
2
a
2
,
1
), we can write Equation 6 as
det (
A

λ
I
) =
λ
2

tr(
A
)
λ
+ det(
A
)
.
(7)
This means that to understand the eigenvalues of all 2
×
2 matrices we only need to consider two
parameters, the trace and the determinant. Suppose that
τ
= tr(
A
) and
Δ
= det(
A
). Now let
B
=
0

1
Δ
τ
.
Since tr(
B
) =
τ
and det(
B
) =
Δ
,
A
and
B
have the same eigenvalues.
This means that to
understand the eigenvalues for 2
×
2 matrices it is su
ffi
cient to only consider matrices of the form
of
B
.
Keep in mind that while
A
and
B
have the same eigenvalues, they will generally have
di
ff
erent eigenvectors.
However, it turns out that we can understand the qualitative behavior
without knowing the eigenvectors. We should also note that the nullclines for the system
d
dt
u
(
t
) =
Bu
(
t
) =
0

1
Δ
τ
x
(
t
)
y
(
t
)
(8)
are the
x
axis and the straight line defined by
y
=

Δ
x/
τ
.
Now suppose that the eigenvalues of a 2
×
2 matrix,
A
are
λ
1
and
λ
2
. Then we must have
det (
A

λ
I
) = (
λ

λ
1
)(
λ

λ
2
) =
λ
2

(
λ
1
+
λ
2
)
λ
+ (
λ
1
λ
2
)
.
(9)
Thus
tr(
A
) = (
λ
1
+
λ
2
) and det(
A
) = (
λ
1
λ
2
)
.
This means that the trace of a 2
×
2 matrix is the sum of the eigenvalues, and its determinant is the
product of the eigenvalues. It turns out that the same argument can be generalized to show that
for any
n
×
n
matrix, the trace of the matrix is the sum of the eigenvalues and the determinant of
the matrix is the product of the eigenvalues.
6